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  • Calculate the limit of a function as x increases or decreases without bound.
  • Recognize a horizontal asymptote on the graph of a function.
  • Estimate the end behavior of a function as x increases or decreases without bound.
  • Recognize an oblique asymptote on the graph of a function.
  • Analyze a function and its derivatives to draw its graph.

We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x ± . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f .

Limits at infinity

We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs , we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

Limits at infinity and horizontal asymptotes

Recall that lim x a f ( x ) = L means f ( x ) becomes arbitrarily close to L as long as x is sufficiently close to a . We can extend this idea to limits at infinity. For example, consider the function f ( x ) = 2 + 1 x . As can be seen graphically in [link] and numerically in [link] , as the values of x get larger, the values of f ( x ) approach 2 . We say the limit as x approaches of f ( x ) is 2 and write lim x f ( x ) = 2 . Similarly, for x < 0 , as the values | x | get larger, the values of f ( x ) approaches 2 . We say the limit as x approaches of f ( x ) is 2 and write lim x a f ( x ) = 2 .

The function f(x) 2 + 1/x is graphed. The function starts negative near y = 2 but then decreases to −∞ near x = 0. The function then decreases from ∞ near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.
The function approaches the asymptote y = 2 as x approaches ± .
Values of a function f As x ±
x 10 100 1,000 10,000
2 + 1 x 2.1 2.01 2.001 2.0001
x −10 −100 −1000 −10,000
2 + 1 x 1.9 1.99 1.999 1.9999

More generally, for any function f , we say the limit as x of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x is sufficiently large. In that case, we write lim x a f ( x ) = L . Similarly, we say the limit as x of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x < 0 and | x | is sufficiently large. In that case, we write lim x f ( x ) = L . We now look at the definition of a function having a limit at infinity.

Definition

(Informal) If the values of f ( x ) become arbitrarily close to L as x becomes sufficiently large, we say the function f has a limit at infinity    and write

lim x f ( x ) = L .

If the values of f ( x ) becomes arbitrarily close to L for x < 0 as | x | becomes sufficiently large, we say that the function f has a limit at negative infinity and write

lim x f ( x ) = L .

If the values f ( x ) are getting arbitrarily close to some finite value L as x or x , the graph of f approaches the line y = L . In that case, the line y = L is a horizontal asymptote of f ( [link] ). For example, for the function f ( x ) = 1 x , since lim x f ( x ) = 0 , the line y = 0 is a horizontal asymptote of f ( x ) = 1 x .

Definition

If lim x f ( x ) = L or lim x f ( x ) = L , we say the line y = L is a horizontal asymptote    of f .

The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.
(a) As x , the values of f are getting arbitrarily close to L . The line y = L is a horizontal asymptote of f . (b) As x , the values of f are getting arbitrarily close to M . The line y = M is a horizontal asymptote of f .

Questions & Answers

differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
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Lambiv
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Venny Reply
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Eliyee
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WARKISA
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Lambiv
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Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
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Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
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Shukri
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Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
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Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
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Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
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Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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